TITLE: How do I know when to use the Law of total probability?
QUESTION [8 upvotes]: I am studying Probability Theory 1, and we have learned the Law of total probability and proved it. I know the theoretical intuition behind it and I know why it makes sense from it's proof. But when I see a new problem (that they solved it using this method) I just can't relate it to this law. 
Can you guide me of how and when should I use it, and what kind of problems can be solved using this theorem? I would like some simple example illustrating it's usage.

REPLY [2 votes]: The examples given by the other guys clearly illustrate the use of the law.  To put it more informally (and perhaps more simply), just check if the main experiment in the problem naturally breaks down into a sequence of sub-experiments.  Usually in the elementary exercises in probability texts, there would be 2 such sub-experiments, one followed by the other.  You would have to compute the probabilities in each sub-experiment and then put them together using the Law of Total Probability to get the answer.  The probability tree is a formally precise description that can be used for such a multi-stage experiment.